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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Lichnerowicz-Formel</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>In der <a href="Mathematik" title="Mathematik">Mathematik</a>, insbesondere im Bereich der <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a>, setzt die <b>Lichnerowicz-Formel</b> den Spinor-Laplace-Operator mit dem Quadrat des <a href="Dirac-Operator" title="Dirac-Operator">Dirac-Operators</a> in Beziehung. Sie ist ein Beispiel einer Weitzenböck-Formel. Benannt ist sie nach <a href="Andr%C3%A9_Lichnerowicz" title="André Lichnerowicz">André Lichnerowicz</a>.
</p><p>Es sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,g)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle (M,g)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e27d2e539fd0c3a9a7efab6257abd17de7fc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle (M,g)}" loading="lazy"></span> eine <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">Riemannsche Mannigfaltigkeit</a> mit Skalarkrümmung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>, und es sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to M}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>M</mi>
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<annotation encoding="application/x-tex">{\displaystyle S\to M}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24ad5efc5478da62b83c74039650dfac91d162dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.556ex; height:2.176ex;" alt="{\displaystyle S\to M}" loading="lazy"></span> ein <a href="Spinorb%C3%BCndel" title="Spinorbündel">Spinorbündel</a> zu einer <a href="Spinstruktur" class="mw-redirect" title="Spinstruktur">Spinstruktur</a> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,g)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (M,g)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e27d2e539fd0c3a9a7efab6257abd17de7fc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle (M,g)}" loading="lazy"></span>, mit <a href="Dirac-Operator" title="Dirac-Operator">Dirac-Operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> und Spinor-Laplace-Operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{S}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msup>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta ^{S}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcb9b032fa1590f0a05907477b146697a29a1875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.228ex; height:2.676ex;" alt="{\displaystyle \Delta ^{S}}" loading="lazy"></span>. Dann besagt die <i>Lichnerowicz-Formel</i>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{2}=\Delta ^{S}+{\frac {R}{4}}Id}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>=</mo>
<msup>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msup>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>R</mi>
<mn>4</mn>
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<mi>I</mi>
<mi>d</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle D^{2}=\Delta ^{S}+{\frac {R}{4}}Id}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8de30b8866aa52b570c8b161a35a8a74b6ffb351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.133ex; height:5.176ex;" alt="{\displaystyle D^{2}=\Delta ^{S}+{\frac {R}{4}}Id}" loading="lazy"></span>.</dd></dl>
<p>Aus der Lichnerowicz-Formel folgt, dass jeder <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</a> des Dirac-Operators die Ungleichung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{2}\geq {\tfrac {1}{4}}{\text{min}}_{x\in M}R(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>≥<!-- ≥ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>4</mn>
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</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mtext>min</mtext>
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<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>M</mi>
</mrow>
</msub>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda ^{2}\geq {\tfrac {1}{4}}{\text{min}}_{x\in M}R(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a65972460c3390dfb5979eee1a80ebb9ab81356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.94ex; height:3.509ex;" alt="{\displaystyle \lambda ^{2}\geq {\tfrac {1}{4}}{\text{min}}_{x\in M}R(x)}" loading="lazy"></span> erfüllt. Insbesondere kann es auf Mannigfaltigkeiten positiver <a href="Skalarkr%C3%BCmmung" class="mw-redirect" title="Skalarkrümmung">Skalarkrümmung</a> keine Spinoren im <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> des Dirac-Operators geben, woraus mit dem <a href="Atiyah-Singer-Indexsatz" title="Atiyah-Singer-Indexsatz">Atiyah-Singer-Indexsatz</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {A}} (TM)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold">A</mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {A}} (TM)=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06a52a5d438a4ebb71806e6412cd408113cb3f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.168ex; height:3.343ex;" alt="{\displaystyle \mathbf {\hat {A}} (TM)=0}" loading="lazy"></span> folgt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>A. Lichnerowicz, "Spineurs harmoniques", C. R. Acad. Sci. Paris, 257: 7–9, 1963</li>
<li>B. H. Lawson, M.-L. Michelsohn: "Spin Geometry", Princeton University Press, 1989, ISBN 978-0-691-08542-5</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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